Astron. Astrophys. 335, 1025-1028 (1998)

2. The Zeeman splitting in CCH

Experimental studies have established beyond doubt that CCH is a linear molecule and has a ground state (Graham et al. 1974; Sastry et al. 1981; Gottlieb et al. 1983). From both laboratory and astrophysical data it has been established that the angular momentum coupling scheme of CCH follows Hund's case (Townes & Schawlow 1975) to a reasonable approximation (Sastry et al. 1981; Ziurys et al. 1982; Gottlieb et al. 1983). Accordingly, we have

where is the rotational angular momentum, the electron spin, the angular momentum and the spin of the hydrogen nucleus. Yet, as indicated by Ziurys et al. (1982), due to the mixing of J-states by the hyperfine interaction, J is no longer a good quantum number, and the usual selection rules no longer hold. Rather, the selection rules for electric dipole transitions now are

Fig. 1 shows the allowed transitions between rotational levels and N, with fine and hyperfine splittings included.

Fig. 1. The energy level diagram of CCH showing all allowed transitions. (Mixing of the J energy levels causes the usual selection rule to break down, hence the transition .) When , the lower rotational level has only one fine structure J level, viz. .

The magnetic fields prevailing in the interstellar medium are so weak that they do not decouple the angular momentum and the nuclear spin . Their effect is just to lift the degeneracy of the hyperfine levels F by splitting them into levels, labeled by the quantum number (). The Zeeman interaction energy is usually written as

where is the Bohr magneton, B the magnetic field, and a dimensionless coefficient known as the Landé factor. The frequency of the emitted radiation for a transition between Zeeman sublevels and is given by

where is the frequency of the transition without magnetic field, , and the index F corresponds to the lower level.

The selection rules for transitions between the sublevels of the rotational levels and N are

From the expressions given by Gordy & Cook (1970) it is easily established that, for the values of the rotational levels we are considering in this paper (), the Landé factor can be written as

Under interstellar conditions, is much less than the full line width at half-maximum (FWHM), so that the individual transitions cannot be resolved, making the Zeeman effect very difficult to detect. However, among all these transitions, the so-called transitions (corresponding to ) are elliptically polarized, with the circularly polarized portion due to the line-of-sight component of the magnetic field. By substracting the left from the right circular polarization of the radio signal, the amplitude of the obtained Stokes V spectrum is proportional to the strength of the line-of-sight component of the magnetic field and to the average frequency separation between the oppositely polarized components (Verschuur 1969; Troland & Heiles 1982). The average is obtained by weighing the frequency separation of the individual pairs of transitions by their relative intensity. The relevant expressions for the latter are (Gordy & Cook 1970)

where F is the smaller of the two F's involved in the transition and the upper sign is taken throughout for the transition and the lower sign for the transition. The coefficients P and Q are independent of .

Tables 1 to 5 give the resulting average frequency separation of the left and right circularly polarized portions of the transitions for the transitions listed in Ziurys et al. (1982).

Table 1. Separation, , of the Zeeman -components for the transition of CCH

Table 2. Separation, , of the Zeeman -components for the transition of CCH

Table 3. Separation, , of the Zeeman -components for the transition of CCH

Table 4. Separation, , of the Zeeman -components for the transition of CCH

Table 5. Separation, , of the Zeeman -components for the transition of CCH

© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
helpdesk.link@springer.de